Since \(X_A\)  is a quasi-smooth hypersurface and of dimension 3, it is known (cf. [1]}) to be in general position relative to \(\mathbb {P}^4(Q)_{\rm sing}\)  (i.e., \(\operatorname{codim}_{X_A} \big (X_A\cap \mathbb {P}^4(Q)_{\rm sing}\big )\ge 2\) , where \(\mathbb {P}^4(Q)_{\rm sing}\)  is the singular locus of \(\mathbb {P}^4(Q)\) ). Hence the dualizing sheaf of \(X_A\)  is computed as \(\omega _{X_A} \cong \mathcal {O}_{X_A} (m-q_0-q_1-q_2-q_3-q_4)\cong \mathcal {O}_{X_A}\)  (see [2]}). The existence of a crepant resolution for \(X_A\)  is proved in [3]}.
